Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 6 x + 31 x^{2} )^{2}$ |
| $1 - 12 x + 98 x^{2} - 372 x^{3} + 961 x^{4}$ | |
| Frobenius angles: | $\pm0.318871840175$, $\pm0.318871840175$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $8$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $676$ | $976144$ | $908057956$ | $855195853824$ | $819449364552676$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $1014$ | $30476$ | $926014$ | $28622900$ | $887388918$ | $27512119340$ | $852891626494$ | $26439641032916$ | $819628381953654$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=11 x^6+5 x^5+10 x^4+15 x^3+16 x^2+4 x+24$
- $y^2=22 x^6+9 x^5+10 x^4+29 x^3+2 x^2+14 x+22$
- $y^2=7 x^6+28 x^5+30 x^4+16 x^3+16 x^2+10 x+13$
- $y^2=23 x^6+15 x^5+11 x^4+19 x^3+11 x^2+15 x+23$
- $y^2=25 x^6+23 x^4+23 x^2+25$
- $y^2=25 x^6+3 x^5+22 x^4+3 x^3+2 x^2+23 x+11$
- $y^2=14 x^6+12 x^5+11 x^4+18 x^3+4 x^2+26 x+27$
- $y^2=27 x^6+29 x^5+19 x^4+9 x^3+26 x^2+11 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$| The isogeny class factors as 1.31.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$ |
Base change
This is a primitive isogeny class.